Advertisement

The mathematics of F. J. Almgren, Jr.

  • Brian White
Article

Abstract

Frederick Justin Almgren, Jr, one of the world’s leading geometric analysts and a pioneer in the geometric calculus of variations, died on February 5, 1997 at the age of 63 as a result of myelodysplasia. Throughout his career, Almgren brought great geometric insight, technical power, and relentless determination to bear on a series of the most important and difficult problems in his field. He solved many of them and, in the process, discovered ideas which turned out to be useful for many other problems. This article is a more-or-less chronological survey of Almgren’s mathematical research. (Excerpts from this article appeared in the December 1997 issue of theNotices of the American Mathematical Society.) Almgren was also an outstanding educator, and he supervised the thesis work of nineteen PhD students; the 1997 volume 6 issue of the journalExperimental Mathematics is dedicated to Almgren and contains reminiscences by two of his PhD students and by various colleagues. A general article about Almgren’s life appeared in the October 1997Notices of the American Mathematical Society [MD]. See [T3]for a brief biography.

Keywords

Minimal Surface Multivalued Function Isoperimetric Inequality Tangent Cone Soap Film 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Papers of F. J. Almgren, Jr

  1. [1]
    F. J. Almgren, Jr The homotopy groups of the integral cycle groups,Topology,1, 257–299, (1962).MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    F. J. Almgren, Jr An isoperimetric inequality,Proc. Am. Math. Soc,15, 284–285, (1964).MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    F. J. Almgren, Jr Three theorems on manifolds with bounded mean curvature,Bull. Am. Math. Soc.,71, 755–756, (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F. J. Almgren, Jr Mass continuous cochains are differential forms,Proc. Am. Math. Soc.,16, 1291–1294, (1965).MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    F. J. Almgren, JrThe Theory of Varifolds. A variational calculus in the large for the k-dimensional are integrand, Multilithed notes, Princeton University Library, 178, 1965.Google Scholar
  6. [6]
    F. J. Almgren, JrPlateau’s Problem. An Invitation to Varifold Geometry. Benjamin, W.A., Ed. New York, 1966.Google Scholar
  7. [7]
    F. J. Almgren, Jr Some interior regularity theorems for minimal surfaces and an extension of Bernstein’s theorem,Ann. Math.,84, 277–292, (1966).MathSciNetCrossRefGoogle Scholar
  8. [8]
    F. J. Almgren, Jr Existence and regularity of solutions to elliptic calculus of variations problems among surfaces of varying topological type and singularity structure,Bull. Am. Math. Soc,73, 576–680, (1967).MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems among surfaces of varying topological type and singularity structure,Ann. Math.,87, 321–391, (1968).MathSciNetCrossRefGoogle Scholar
  10. [10]
    F. J. Almgren, Jr Measure theoretic geometry and elliptic variational problems,Bull. Am. Math. Soc,75, 285–304, (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    F. J. Almgren, Jr A maximum principle for elliptic variational problems,J. Functional Anal.,4, 380–389, (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    F. J. Almgren, Jr Measure theoretic geometry and elliptic variational problems,Proceedings of the Symposium on Continuum Mechanics and Related Problems of Analysis, (Tbilisi, 1971), (in Russian) vol.II, 307–324. Izdat. “Mecniereba,” Tbilisi, 1974. fMathSciNetGoogle Scholar
  13. [13]
    F. J. Almgren, Jr Geometric measure theory and elliptic variational problems,Actes du Congrès International des Mathématiciens, (Nice, 1970), Tome 2, Gauthier-Villars, Paris, 813–819, 1971.Google Scholar
  14. [14]
    F. J. Almgren, Jr with Allard, W.K. An introduction to regularity theory for parametric elliptic variational problems. Partial differential equations,Proc. Symp. Pure Math.,XXIII, 231–260, 1973;Am. Math. Soc., Providence, RI.MathSciNetGoogle Scholar
  15. [15]
    F. J. Almgren, Jr Geometric variational problems from a measure-theoretic point of view,Global analysis and its applications, (Lectures, Internat. Sem. Course, Internat. Centre Theoret. Phys., Trieste, 1972), Vol.II, Internat. Atomic Energy Agency, Viena, 1–22, 1974.Google Scholar
  16. [16]
    F. J. Almgren, Jr Geometric measure theory and elliptic variational problems.Geometric Measure Theory and Minimal Surfaces, (C.I.M.E. Lectures, III Ciclo, Varenna, 1972), Ediziono Cremonese, Rome, 31–117, 1973.Google Scholar
  17. [17]
    F. J. Almgren, Jr The structure of limit varifolds associated with minimizing sequences of mappings,Symposia Mathematica,XIV. (Convegno di Teoria Geometrica dell’Integrazione e Varietá Minimali, INDAM, Rome, 1973), Academic Press, London, 1974.Google Scholar
  18. [18]
    F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Bull. Am. Math. Soc.,81, 151–154, (1975).MathSciNetCrossRefzbMATHGoogle Scholar
  19. [19]
    F. J. Almgren, Jr Existence and regularity almost everywhere of solutions to elliptic variational problems with constraints,Mem. Am. Math. Soc.,4(165), viii + 199, (1976).MathSciNetGoogle Scholar
  20. [20]
    F. J. Almgren, Jr with Allard, W.K. The structure of stationary one-dimensional varifolds with positive density,Inv. Math.,34, 83–97, (1976).MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    F. J. Almgren, Jr with Taylor, J.E. The geometry of soap films and soap bubbles,Sci. Am., 82–93, July (1976).Google Scholar
  22. [22]
    F. J. Almgren, Jr with Thurston, W.P. Examples of unknotted curves which bound only surfaces’ of high genus with their convex hulls,Ann. Math.,105, 527–538, (1977).MathSciNetCrossRefGoogle Scholar
  23. [23]
    F. J. Almgren, Jr with Schoen, R. and Simon, L. Regularity and singularity estimates of hypersurfaces minimizing parametric elliptic variational integrals,Acta Math.,139, 217–265, (1977).MathSciNetCrossRefGoogle Scholar
  24. [24]
    F. J. Almgren, Jr with Simon, L. Existence of embedded solutions of Plateau’s problem,Annali Scuola Normale Superiore de Pisa (Series IV),VI(3), 447–495, (1979).MathSciNetGoogle Scholar
  25. [25]
    F. J. Almgren, Jr Dirichlet’s problem for multiple valued functions and the regularity of mass minimizing integral currents, Minimal submanifolds and geodesics, Obata, M., Ed.,Proceedings of the Japan-U.S. Seminar on Minimal Submanifolds including Geodesies, Kaigai Publishings, Tokyo, Japan, 1–6, 1978.Google Scholar
  26. [26]
    F. J. Almgren, Jr with Taylor, J.E. Descriptive geometry in the calculus of variations, Proceedings of the International Congress on Descriptive Geometry (Vancouver, 1978),Engineering Design Graphics J., (1979).Google Scholar
  27. [27]
    F. J. Almgren, Jr Minimal surfaces: tangent cones, singularities, and topological types.Proceedings of the International Congress of Mathematicians (Helsinki, 1978), Lehto, O. Ed., Acad. Sci. Fennica, Helsinki,2, 767–770, 1980.Google Scholar
  28. [28]
    F. J. Almgren, Jr with Allard, W.K. On the radial behavior of minimal surfaces and the uniqueness of their tangent cones,Ann. Math.,113, 215–265, (1981).MathSciNetCrossRefGoogle Scholar
  29. [29]
    F. J. Almgren, Jr with Thurston, R.N. Liquid crystals and geodesies,J. Phys.,42, 413–417, (1981).MathSciNetGoogle Scholar
  30. [30]
    F. J. Almgren, JrMinimal Surfaces. McGraw-Hill Encyclopedia of Science and Technology, 5th ed., McGraw-Hill, New York, 599–600, 1982.Google Scholar
  31. [31]
    F. J. Almgren, Jr Minimal surface forms,The Mathematical Intelligencer,4(4), 164–172, (1982).MathSciNetzbMATHGoogle Scholar
  32. [32]
    F. J. Almgren, Jr Approximation of rectifiable currents by Lipschitz Q-valued functions, Seminar on Minimal Submanifolds,Ann. Math. Studies, Princeton University Press, Princeton, NJ,103, 243–259, (1983).Google Scholar
  33. [33]
    F. J. Almgren, Jr Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two,Bull. Am. Math. Soc.,8(2), 327–328, (1983).MathSciNetCrossRefGoogle Scholar
  34. [34]
    F. J. Almgren, Jr Q-valued functions minimizing Dirichlet’s integral and the regularity of area minimizing rectifiable currents up to codimension two, (V. Scheffer and J. Taylor, Eds.),World Scientific, to appear. Currently available electronically at http://www.math.princeton.edu/~scheffer.Google Scholar
  35. [35]
    F. J. Almgren, Jr with Super, B. Multiple valued functions in the geometric calculus of variations,Astérisque,118, 13–32, (1984).MathSciNetGoogle Scholar
  36. [36]
    F. J. Almgren, Jr Optimal isoperimetric inequalities,Bull. Am. Math. Soc,13(2), 123–126, (1985).MathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    F. J. Almgren, Jr Optimal isoperimetric inequalities,Indiana V. Math. J.,35(3), 451–547, (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    F. J. Almgren, Jr Deformations and multiple valued functions, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984),Proc. Sympos. Pure Math., Am. Math. Soc., Providence, RI,44, 29–130, (1986).MathSciNetGoogle Scholar
  39. [39]
    F. J. Almgren, Jr Applications of multiple valued functions,Geometric Modeling: Algorithms and New Trends, SIAM, Philadelphia, 43–54, 1987.Google Scholar
  40. [40]
    F. J. Almgren, Jr Spherical symmetrization,Proceedings of the International Workshop on Integral Functions in the Calculus of Variations (Trieste, 1985), Rend. Circ. Mat. Palermo (2) Suppl., 11–25, 1987.Google Scholar
  41. [41]
    F. J. Almgren, Jr with Taylor, J.E. Optimal crystal shapes,Variational Methods for Free Surface Interfaces, Concus, P. and Finn, R., Eds., Springer-Verlag, New York, 1–11, 1987.Google Scholar
  42. [42]
    F. J. Almgren, Jr with Lieb, E.H. Singularities of energy minimizing maps from the ball to the sphere,Bull. Am. Math. Soc.,17, 304–306, (1987).MathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    F. J. Almgren, Jr with Browder, W. and Lieb, E.H. Co-area, liquid crystals, and minimal surfaces,Partial Differential Equations, Chern, S.S., Ed., Springer Lecture Notes in Mathematics 1306, Springer-Verlag, New York, 1–22, 1988.CrossRefGoogle Scholar
  44. [44]
    F. J. Almgren, Jr with Lieb, E.H. Singularities of energy minimizing maps from the ball to the sphere,Ann. Math.,128, 483–530, (1988).MathSciNetCrossRefGoogle Scholar
  45. [45]
    F. J. Almgren, Jr with Lieb, E.H. Counting singularities in liquid crystals,Symposia Mathematica, Vol. XXX (Cortona, 1988), Academic Press, London, 103–118, 1989; also in:IXth International Congress on Mathematical Physics, (Swansea, 1988), Hilger, Bristol, 396–409, 1989; also in: Variational methods, (Paris, 1988), 17–35,Progr. Nonlinear Differential Equations Appl.,4, Birkhäuser, Boston, MA, 1990. also (under the title “How many singularities can there be in an energy minimizing map from the ball to the sphere?”) in: Ideas and methods in mathematical analysis, stochastics, and applications, Cambridge University Press, Cambridge, MA, Albeverio, S., Fenstad, J.E., Holden, H., and Lindstrom, T., Eds., 394–408, 1992.Google Scholar
  46. [46]
    F. J. Almgren, Jr with Lieb, E.H. Symmetric decreasing rearrangement can be discontinuous,Bull. Am. Math. Soc.,20, 177–180, (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    F. J. Almgren, Jr with Lieb, E.H. Symmetric rearrangement is sometimes continuous,J. Am. Math. Soc.,2, 683–773, (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    F. J. Almgren, Jr with Gurtin, M. A mathematical contribution to Gibbs’s analyses of fluid phases in equilibriumPartial Differential Equations and the Calculus of Variations, Progr. Nonlinear Differential Equations Appl., Birkäuser, Boston,1, 9–28, 1989.Google Scholar
  49. [49]
    F. J. Almgren, Jr with Browder, W. Homotopy with holes and minimal surfaces,Differential Geometry. Lawson, B. and Tenenblat, K., Eds., Pitman Monographs Surveys Pure Appl. Math., Longman Scientific & Technical, Harlow,52, 15–23, 1991.Google Scholar
  50. [50]
    F. J. Almgren, Jr What can geometric measure theory do for several complex variables? Proceedings of the Several Complex Variables Year at the Mittag-Leffler Institute (Stockholm, 1987–1988), Princeton University Press Math. Notes (38), Princeton, NJ, 8–21, 1993.Google Scholar
  51. [51]
    F. J. Almgren, Jr The geometric calculus of variations and modelling natural phenomena, Statistical thermodynamics and differential geometry of microstructured materials (Minneapolis, MN, 1991),IMA Vol. Math. Appl., Springer-Verlag, New York,51, 1–5, (1993).Google Scholar
  52. [52]
    F. J. Almgren, Jr Multi-functions modv, Geometric analysis and computer graphics (Berkeley, CA, 1988), 1–17,Math. Sci. Res. Inst. Publ., Springer-Verlag, New York,17, (1991).Google Scholar
  53. [53]
    F. J. Almgren, Jr with Lieb, E.H. The (non)continuity of symmetric decreasing rearrangement, Proceedings of the conference on geometry of solutions to PDE (Cortona, 1988),Symposia Mathematica, Academic Press, Boston, MA, XXX, 1992; also in: Variational methods (Paris, 1988),Progr. Nonlinear Differential Equations Appl, Birkhäuser, Boston, MA, 4,3–16,1990. also in: Differential equations and mathematical physics, (Birmingham, AL, 1990),Math. Sci. Engrg., Academic Press, Boston, MA,186, 183–200, 1992.Google Scholar
  54. [54]
    F. J. Almgren, Jr Computing soap films and crystals,Computing Optimal Geometries, video report,Am. Math. Soc., 1991.Google Scholar
  55. [55]
    F. J. Almgren, Jr with Sullivan, J. Visualization of soap bubble geometries,Leonardo,25, 267–271, (1992).CrossRefGoogle Scholar
  56. [56]
    F. J. Almgren, Jr with Taylor, J.E. and Wang, L. A variational approach to motion by weighted mean curvature, Computational Crystal Growers Workshop Selected Lectures in Mathematics,Am. Math. Soc., 9–12, (1992).Google Scholar
  57. [57]
    F. J. Almgren, Jr with Taylor, J.E. and Wang, L. Curvature driven flows: A variational approach,S1AM J. Control and Optimization,31(2), 387–438, (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  58. [58]
    F. J. Almgren, Jr Questions and answers about area minimizing surfaces and geometric measure theory, Differential Geometry: partial differential equations on manifolds, (Los Angeles, 1990),Proc. Symposia Pure Math., Am. Math. Soc,51, 29–53, 1992.Google Scholar
  59. [59]
    F. J. Almgren, Jr with Taylor, J.E. Flat flow is motion by crystalline curvature for curves with crystalline energies,J. Differential Geom.,42(1), 1–22, (1995).MathSciNetzbMATHGoogle Scholar
  60. [60]
    F. J. Almgren, Jr with Taylor, J.E. Optimal geometry in equilibrium and growth, Symposium in Honor of Benoit Mandelbrot (Curaçao, 1995),Fractals,3(4), 713–723, (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  61. [61]
    F. J. Almgren, Jr Questions and answers about geometric evolution processes and crystal growth,The Gelfand Mathematical Seminars, 1–9, 1993–1995; Gelfand Math. Sem., Birkhäuser, Boston, 1996.Google Scholar
  62. [62]
    F. J. Almgren, Jr with Wang, L. Mathematical existence of crystal growth with Gibbs-Thomson curvature effects,J. Geom. Anal, (to appear).Google Scholar
  63. [63]
    F. J. Almgren, Jr with Rivin, I. The mean curvature integral is invariant under bending, 1–21, Geometry and topology monographs #1, University of Warwick published electronically: www.maths.warwick.ac.uk/gt/main/mlGoogle Scholar
  64. [64]
    F. J. Almgren, Jr with Taylor, J. Soap bubble clusters: the Kelvin problem,Forma,11(3), 199–207, (1996).MathSciNetzbMATHGoogle Scholar
  65. [65]
    F. J. Almgren, JrGlobal Analysis. preprint, (survey/expository).Google Scholar
  66. [66]
    F. J. Almgren, Jr Isoperimetric inequalities for anisotropic surface energies, unfinished manuscript.Google Scholar
  67. [67]
    F. J. Almgren, Jr A new look at flat chains modn, unfinished manuscript.Google Scholar

References

  1. [AdS]
    Adams, D. and Simon, L. Rates of asymptotic convergence near isolated singularities of geometric extrema,Indiana U. Math. J.,37, 225–254, (1988).MathSciNetCrossRefzbMATHGoogle Scholar
  2. [Al1]
    Allard, W.K. On the first variation of a varifold,Annal. Math.,95, 417–491, (1972).MathSciNetCrossRefGoogle Scholar
  3. [Al2]
    Allard, W.K. An integrality theorem and a regularity theorem for surfaces whose first variation with respect to a parametric elliptic integrand is controlled,Am. Math. Soc. Proc. Symp. Pure Math., (Arcata),44, 1–28, (1986).MathSciNetGoogle Scholar
  4. [AmB]
    Ambrosio, L. and Braides, A. Functionals defined on partitions in sets of finite perimeter I, II,J. Math. Pures Appl.,69, 285–333, (1990).MathSciNetzbMATHGoogle Scholar
  5. [Anz]
    Anzellotti, G. On the c1,α -regularity of ω-minima of quadratic functionals,Boll. Un. Math. Ital. C (6) 2(1), 195–212, (1983).MathSciNetzbMATHGoogle Scholar
  6. [BDG]
    Bombieri, E., De Giorgi, E., and Giusti, E. Minimal cones and the Bernstein problem,Invent. Math.,7, 243–268, (1969).MathSciNetCrossRefzbMATHGoogle Scholar
  7. [Bo]
    Bombieri, E. Regularity theory for almost minimal currents,Arch. Rational Mech. Anal,78, 99–130, (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  8. [Br1]
    Brakke, K.The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton, NJ, 1977.Google Scholar
  9. [Br2]
    Brakke, K. The Surface Evolver,Experimental Mathematics,1, 141–165, 1992.MathSciNetzbMATHGoogle Scholar
  10. [Br3]
    Brakke, K. Soap films and covering spaces,J. Geom. Anal,5, 445–514, (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  11. [Ca]
    Calabi, E. Minimal immmersions of surfaces in Euclidean spheres,J. Diff. Geometry,1, 111–125, (1967).MathSciNetzbMATHGoogle Scholar
  12. [CE]
    Cao, J. and Escobar, J.F. An isoperimetric comparison theorem for PL-manifolds of non-positive curvature, preprint, (revised version), 1998.Google Scholar
  13. [Cha1]
    Chang, S. Two-dimensional area minimizing integral currents are classical minimal surfaces,J. Am. Math. Soc,1, 699–778, (1988).CrossRefzbMATHGoogle Scholar
  14. [Cho1]
    Choe, J. The isoperimetric inequality for a minimal surface with radially connected boundary,Ann. ScuolaNorm. Sup. Pisa Cl. Sci.,17(4), 583–593, (1990).MathSciNetzbMATHGoogle Scholar
  15. [Cho2]
    Choe, J. Three sharp isoperimetric inequalities for stationary varifolds and area minimizing flat chains modk, Kodai Math. J.,19, 177–190, (1996).MathSciNetCrossRefzbMATHGoogle Scholar
  16. [C,r]
    Croke, C. A sharp four-dimensional isoperimetric inequality,Comment. Math. Helv.,59(2), 187–192, (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  17. [DP]
    De Pauw, T. On SBV dual,Indiana U. Math. J., bf47, 99–121, (1998).CrossRefzbMATHGoogle Scholar
  18. [E]
    Evans, L.C. Quasiconvexity and partial regularity in the calculus of variations,Arch. Rational Mech. Anal,95, 227–252.Google Scholar
  19. [F]
    Federer, H. Flat chains with positive densities,Indiana U. Math. J.,35, 413–424, (1986).MathSciNetCrossRefzbMATHGoogle Scholar
  20. [FF]
    Federer, H. and Fleming, W. Normal and integral currents,Ann. Math.,72, 458–520, (1960).MathSciNetCrossRefGoogle Scholar
  21. [GaL]
    Garafolo, N. and Lin, F.-H. Monotonicity properties of variational integrals,A p weights unique continuation,Indiana U. Math. J.,35, 245–267, (1986).CrossRefGoogle Scholar
  22. [GiG]
    Giaquinta, M. and Giusti, E. Quasiminima,Ann. Inst. H. Poincaré Anal. Non Linéare,1, 79–107, (1984).MathSciNetzbMATHGoogle Scholar
  23. [GMS]
    Giaquinta, M., Modica, G., and Souček, J. Cartesian currents, weak diffeomorphisms, and existence theorems in nonlinear elasticity,Arch. Rational Mech. Anal,106, 97–159, (1989); Correction,109, 385–392, (1990).MathSciNetCrossRefzbMATHGoogle Scholar
  24. [GMS2]
    Giaquinta, M., Modica, G., and Souček, J. Cartesian currents in the calculus of variations, 2 volumes, Springer-Verlag, 1998.Google Scholar
  25. [GS]
    Gromov, M. and Schoen, R. Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one,Inst. Hautes Etudes Sci. Publ. Math.,76, 165–246, (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  26. [GW]
    Gulliver, R. and White, B. The rate of convergence of a harmonic map at a singular point,Math. Ann.,283, 539–549, (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  27. [Ham]
    Hamilton, R. Three-manifolds with positive ricci curvature,J. Diff. Geom.,17, 255–306, (1982).MathSciNetzbMATHGoogle Scholar
  28. [Ha,r]
    Hardt, R. On boundary regularity for integral currents or flat chains modulo two minimizing the integral of an elliptic integrand,Communications in P.D.E.,2, 1163–1232, (1977).MathSciNetCrossRefGoogle Scholar
  29. [HL]
    Hardt, R. and Lin, F.H. Harmonic maps into round cones and singularities of nematic liquid crystals,Math. Z.,213, 575–593, (1993).MathSciNetzbMATHCrossRefGoogle Scholar
  30. [HKL]
    Hardt, R., Kinderlehrer, D., and Lin, F.H. Stable defects of minimizers of constrained variational principles,Ann. Inst. H. Poincaré, Anal. Nonl.,5, 297–322, (1988).MathSciNetzbMATHGoogle Scholar
  31. [Hub]
    Hubbard, J.H. On the convex hull genus of space curves,Topology,19, 203–208, (1980).MathSciNetCrossRefzbMATHGoogle Scholar
  32. [Hut]
    Hutchinson, J. C1,α multiple function regularity and tangent cone behaviour for varifolds with mean curvature inL p,Am. Math. Soc. Proc. Symp. Pure Math., (Arcata),44, 281–306, (1986).MathSciNetGoogle Scholar
  33. [K]
    Kleiner, B. An isoperimetric comparison theorem,Invent. Math.,108, 37–47, (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  34. [LSY]
    Li, P., Schoen, R., and Yau, S.-T. On the isoperimetric inequality for minimal surfaces,Ann. Scuola Norm. Sup. Pisa Cl. Sci.,11(4), 237–244, (1984).MathSciNetzbMATHGoogle Scholar
  35. [L]
    Lin, F.-H. Nodal sets of solutions of elliptic and parabolic equations,Comm. Pure Appl. Math.,44, 287–308, (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  36. [Lu]
    Luckhaus, S. Solutions for the two-phase Stefan problem with Gibbs-Thomson law for the melting temperature,Euro. J. App. Math.,1, 101–111, (1990).MathSciNetzbMATHGoogle Scholar
  37. [MD]
    Mackenzie, D. Fred Almgren (1933–1997): lover of mathematics, family, and life’s adventures,Notices Am. Math. Soc,44(9), 1102–1106, (1997).MathSciNetzbMATHGoogle Scholar
  38. [MW]
    Micallef, M. and White, B. The structure of branch points in area minimizing surfaces and in pseudoholomorphic curves,Ann. Math.,141, 35–85, (1995).MathSciNetCrossRefzbMATHGoogle Scholar
  39. [MS]
    Michael, J. and Simon, L. Sobolev and mean-value inequalities on generalized submanifolds of Rn,Comm. Pure Appl. Math.,26, 361–379, (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  40. [MF1]
    Morgan, F. Area minimizing currents bounded by higher multiples of curves,Rend. Circ. Mat. Palermo,33, 37–46, (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  41. [MF2]
    Morgan, F. The cone over the Clifford torus in R4 is Φ-minimizing,Math. Ann.,289, 341–354, (1991).MathSciNetCrossRefzbMATHGoogle Scholar
  42. [MCB]
    Morrey, C.B. Nonlinear elliptic systems,J. Math. Meck,17, 649–670, (1968).MathSciNetzbMATHGoogle Scholar
  43. [O]
    Osserman, R. The isoperimetric inequality,Bull. Am. Math. Soc,84(6), 1182–1238, (1978).MathSciNetCrossRefzbMATHGoogle Scholar
  44. [P]
    Pitts, J.Existence and Regularity of Minimal Surfaces on Riemannian Manifolds, Princeton University Press, Princeton, NJ, 1977.Google Scholar
  45. [SS]
    Schoen, R. and Simon, L. A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals,Indiana U. Math. J.,31, 415–434, (1982).MathSciNetCrossRefzbMATHGoogle Scholar
  46. [SJ]
    Simons, J. Minimal varieties in Riemannian manifolds,Ann. Math.,88, 62–105, (1968).MathSciNetCrossRefGoogle Scholar
  47. [SL1]
    Simon, L. Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems,Ann. Math.,118, 525–571, (1983).CrossRefGoogle Scholar
  48. [SL2]
    Simon, L. Isolated singularities of extrema of geometric variational problems, in Harmonic mappings and minimal immersions (Montecatini, 1984),Springer Lecture Notes in Math.,1161, 206–277.Google Scholar
  49. [SL3]
    Simon, L. Cylindrical tangent cones and the singular set of minimal submanifolds,J. Diff. Geom.,38, 585–652, (1993).zbMATHGoogle Scholar
  50. [SL4]
    Simon, L. Rectifiability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps, inSurveys in Differential Geometry, Vol. II, (Cambridge, MA, 1993), Internat. Press, Cambridge, MA, 246–305, 1995.Google Scholar
  51. [SL5]
    Simon, L. Rectifiability of the singular set of energy minimizing maps,Calc. Var. Part. Diff. Eq.,3(1), 1–65, (1995).zbMATHGoogle Scholar
  52. [SL6]
    Simon, L. Theorems on regularity and singularity of energy minimizing maps, Based on lecture notes by Norbert Hungerbühler, inLectures in Mathematics ETH Zürich, Birkhäuser, Springer-Verlag, Basel, viii + 152, (1996).Google Scholar
  53. [Sol]
    Solomon, B. A new proof of the closure theorem for integral currents,Indiana U. Math. J.,33, 393–418, (1984).CrossRefzbMATHGoogle Scholar
  54. [So2]
    Solomon, B. The harmonic analysis of cubic isoparametric hypersurfaces I, II,Am. J. Math.,112, 157–241, (1990).CrossRefGoogle Scholar
  55. [T1]
    Taylor, J.E. Regularity of the singular sets of two-dimensional area-minimizing flat chains modulo 3 in R3,Invent. Math.,22, 119–159, (1973).MathSciNetCrossRefzbMATHGoogle Scholar
  56. [T2]
    Taylor, J.E. The structure of singularities in soap-bubble-like and soap-film-like minimal surfaces,Ann. Math.,103, 489–539, (1976).CrossRefGoogle Scholar
  57. [T3]
    Taylor, J.E. Frederick Justin Almgren, 1933–1997,Journal of Geometric Analysis,8, 673–674(1998).Google Scholar
  58. [We]
    Weil, A. Sur les surfaces a courbure negative,C. R. Acad. Sci.,182, 1069–1071, (1926); also in:Collected Works,I, Springer-Verlag, New York, 1979.zbMATHGoogle Scholar
  59. [W1]
    White, B. Tangent cones to 2-dimensional area-minimizing integral currents are unique,Duke Math. J.,50, 143–160, (1983).MathSciNetCrossRefzbMATHGoogle Scholar
  60. [W2]
    White, B. The least area bounded by multiples of a curve,Proc. Am. Math. Soc.,90, 230–232, (1984).CrossRefzbMATHGoogle Scholar
  61. [W3]
    White, B. A regularity theorem for minimizing hypersurfaces modp, Geometric Measure Theory and the Calculus of Variations,Am. Math. Soc., Proc. Symp. Pure Math.,44, 413–427, (1986).Google Scholar
  62. [W4]
    White, B. A new proof of the compactness theorem for integral currents,Comm. Math. Helv.,64, 207–220, (1989).CrossRefzbMATHGoogle Scholar
  63. [W5]
    White, B.Some Recent Developments in Differential Geometry, Math. Intelligencer,11, 41–47, 1989.MathSciNetzbMATHGoogle Scholar
  64. [W6]
    White, B. Nonunique tangent maps at isolated singularities of harmonic maps,Bull. Am. Math. Soc,26, 125–129, (1992).CrossRefzbMATHGoogle Scholar
  65. [W7]
    White, B. Existence of least-energy configurations of immiscible fluids,J. Geom. Anal.,6, 151–161, (1996).MathSciNetzbMATHGoogle Scholar
  66. [W8]
    White, B. Statification of minimal surfaces, mean curvature flows, and harmonic maps,J. Reine Ang. Math.,488, 1–35, (1997).zbMATHGoogle Scholar
  67. [W9]
    White, B. Classical area minimizing surfaces with real analytic boundaries,Acta Math.,179, 295–305, (1997).MathSciNetCrossRefzbMATHGoogle Scholar
  68. [W10]
    White, B. The deformation theorem for flat chains,Acta Math., (to appear).Google Scholar
  69. [W11]
    White, B. Rectifiability of flat chains,Annals of Math., (to appear).Google Scholar
  70. [Y1]
    Young, L.C. Generalized surfaces in the calculus of variations I, II,Ann. Math.,43, 84–103, 530–544, (1942).CrossRefGoogle Scholar
  71. [Y2]
    Young, L.C. Surfaces paramétriques généralisées,Bull. Soc. Math. France,79, 59–84, (1951).MathSciNetzbMATHGoogle Scholar
  72. [Y3]
    Young, L.C. Some extremal questions for simplicial complexes, V: the relative area of a Klein bottle,Rend. Circ. Mat. Palermo,12, 257–274, (1963).MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1998

Authors and Affiliations

  1. 1.Mathematics DepartmentStanford UniversityStanford

Personalised recommendations